Sets, Relations, and Integers

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Chapter 1
Sets, Relations, and Integers
The purpose of this introductory chapter is mainly to review briefly some familiar properties of sets, functions, and number theory. Although most of these properties are familiar to the reader, there are certain concepts and results which are basic to the understanding of the body of the text.
This chapter is also used to set down the conventions and notations to be used throughout the book. Sets will always be denoted by capital letters. For example, we use the notation for the set of positive integers, for the set of integers, for the set of nonnegative integers, for the set of even integers, for the set of rational numbers, for the set of positive rational numbers, for the set of nonzero rational numbers, for the set of real numbers, for the set of positive real numbers, for the set of nonzero real numbers, for the set of complex numbers, and for the set of nonzero complex numbers.
1.1 Sets
We will not attempt to give an axiomatic treatment of set theory. Rather we use an intuitive approach to the subject. Consequently, we think of a set as some given collection of objects. A set with only a finite number of elements is called a finite set; otherwise is called an infinite set. We let denote the number of elements of . We quite often denote a finite set by a listing of its elements within braces. For example, is the set consisting of the objects . This technique is sometimes used for infinite sets. For instance, the set of positive integers may be denoted by .
Given a set , we use the notation and to mean is a member of and is not a member of , respectively. For the set , we have and .
A set is said to be a subset of a set if every element of is an element of . In this case, we write and say that is contained in . If , but , then we write and say that is properly contained in or